Find the LCM of the numbers given below :
42, 63

Options

A. 116
B. 126
C. 136
D. 146

Answer

To find the Least Common Multiple (LCM) of the numbers 42 and 63, we can follow these steps:

### Step 1: Find the prime factorization of each number.

**For 42:**
- Divide by 2 (the smallest prime number):
\( 42 \div 2 = 21 \)
So, \( 42 = 2 \times 21 \)

- Now, factor 21:
Divide by 3:
\( 21 \div 3 = 7 \)
So, \( 21 = 3 \times 7 \)

- Therefore, the prime factorization of 42 is:
\( 42 = 2^1 \times 3^1 \times 7^1 \)

**For 63:**
- Divide by 3 (the smallest prime number):
\( 63 \div 3 = 21 \)
So, \( 63 = 3 \times 21 \)

- Now, factor 21 (as we did earlier):
\( 21 = 3 \times 7 \)

- Therefore, the prime factorization of 63 is:
\( 63 = 3^2 \times 7^1 \)

### Step 2: Identify the highest power of each prime factor.

- From the prime factorizations:
- For prime number 2: The highest power is \( 2^1 \) (from 42).
- For prime number 3: The highest power is \( 3^2 \) (from 63).
- For prime number 7: The highest power is \( 7^1 \) (common in both).

### Step 3: Multiply the highest powers of all prime factors together to find the LCM.

LCM = \( 2^1 \times 3^2 \times 7^1 \)

Calculating this step-by-step:
- \( 2^1 = 2 \)
- \( 3^2 = 9 \)
- \( 7^1 = 7 \)

Now, multiply these together:
1. \( 2 \times 9 = 18 \)
2. \( 18 \times 7 = 126 \)

Thus, the LCM of 42 and 63 is **126**.

### Summary of the Steps:
1. Prime factorization of 42: \( 2^1 \times 3^1 \times 7^1 \)
2. Prime factorization of 63: \( 3^2 \times 7^1 \)
3. Highest powers: \( 2^1, 3^2, 7^1 \)
4. LCM = \( 2^1 \times 3^2 \times 7^1 = 126 \)